Bisector of Angle between Two Vectors

Q. The vector$\overrightarrow{V}$ directed along the internal bisector of the angle between the vectors
$\overrightarrow{A}=3\hat{i}-4\hat{j},\overrightarrow{B}=4\hat{i}+2\hat{j}-4\hat{k}$ and $\left(\overrightarrow{V}\right|=2$ is

1. $\frac{38\hat{i}-14\hat{j}-20\hat{k}}{\sqrt{510}}$
2. $\frac{7}{2}\hat{i}-\hat{j}-2\hat{k}$
3. $\frac{14\hat{i}-4\hat{j}-8\hat{k}}{\sqrt{69}}$

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices $A,B$ and $C$, respectively of triangle $ABC$, then the position vector of the point where the bisector of $\angle A$ meets $BC$ is

1. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
2. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{3}(5\hat{j}+12\hat{k})$

Q. The vector $\overrightarrow{C}$, directed along the internal bisector of the angle between the vectors $\overrightarrow{a}=7\hat{i}-4\hat{j}-4\hat{k}$ and $\overrightarrow{b}=-2\hat{i}-\hat{j}+2\hat{k}$ with $|\overrightarrow{c}|=5\sqrt{6}$, is

1. $\pm \left(\frac{5}{3}\right(\hat{i}-7\hat{j}+2\hat{k})$
2. $\frac{5}{3}(5\hat{i}-5\hat{j}+2\hat{k})$
   
3. $\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$
   
4. $\frac{5}{3}(-5\hat{i}-5\hat{j}+2\hat{k})$

Q. The vector $a\hat{i}+b\hat{j}+c\hat{k}$ is a bisector of the angle between the vectors $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ if

1. a=b
2. a=c
3. c=a+b
4. a =b=c